Converter, encryption/decryption system, multi-stage converter, converting method, multi-stage converting method, program, and information recording medium

ABSTRACT

A converter uses a predetermined parameter a. A generating unit accepts generated inputs x 1 , . . . , x n , and generates generated outputs, y 1 , . . . , y n , using recurrence formulas, y 1 =F 1  (x 1 , a) and y i+1 =F i+1  (x i+1 , y 1 ) (1≦i≦n−1). A key accepting unit accepts key inputs, k 1 , . . . , k n , and gives them as generated inputs to said generating unit. A repetition controller gives the generated outputs as generated inputs to said generating unit, for an “m” (m≧0) number of times, and sets one of the generated outputs to be given at the end as a random number string, r 1 , . . . , r n . The data accepting unit accepts data inputs, d 1 , . . . , d n . The converting unit converts data using, e i =d i ⋆r i , and, outputs data outputs, e 1 , . . . , e n . The converter can be used both for encrypting and decrypting data.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.10/233,119, filed Aug. 29, 2002, which is hereby incorporated in itsentirety by reference herein.

FIELD OF THE INVENTION

The present invention relates to a converter, an encryption/decryptionsystem, a multi-stage converter, a converting method, a multi-stageconverting method, a program and an information recording mediumrecording information, which are preferable for a vector-stream privatekey encryption system.

DESCRIPTION OF THE RELATED ART

Conventionally, as a private key encryption system, a block encryptionmethod or a stream encryption method are known. The standard of theblock encryption method includes DES, RC5, etc., and the standard of thestream encryption method includes RC4, SEAL 1.0, etc.

According to the stream encryption method, a random bit string isgenerated, and an exclusive OR operation is applied between target datato be encrypted and this generated random bit string, thereby encryptingthe target data. Hence, the encryption speed depends on the generationspeed of the random bit string, so that the encryption can be realizedgenerally at high speed. The stream encryption method is preferred forthe contents (mobile communications, etc.) wherein bit errors are notnegligible, and realizes flexible change in the data format.

In the block encryption method, non-linear mixing of data, i.e. an “S”function, is used. Data processing is performed in the unit of blocks,it is an advantageous aspect that various data formats (image data,audio data, motion pictures, etc.) can be employed in this encryptionmethod. However, if there is a bit error in the data, the error may bediffused.

It is highly demanded that there should be a private key encryptionsystem having both the advantage of the above-described streamencryption technique and the advantage of the block encryptiontechnique.

In particular, demanded is a private key encryption system which issuitable for encrypting a large volume of data, such as large-scaledatabases, image data, audio data, motion pictures, etc.

SUMMARY OF THE INVENTION

The present invention has been made in consideration of the above. It isaccordingly an object of the present invention to provide a converter,an encryption/decryption system, a multi-stage converter, a convertingmethod, a multi-stage converting method, a program and an informationrecording medium, which are preferable for a vector-stream private keyencryption system.

In order to accomplish the above object, according to the first aspectof the present invention, there is provided a converter using:

an “n” (n≧1) number of conversion functions, F_(i): A×A→A (1≦i≦n), withrespect to a domain A;

a binary arithmetic operation, ⋆: A×A→A, and its reverse binaryarithmetic operation, ⊚: A×A→A, wherein,

-   -   for arbitrary xεA, yεA, conditions of

(x⋆y)⊚y=x, and

(x⊚y)⋆y=x

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converter comprising a generating unit, a key accepting unit, arepetition controller, a data accepting unit, and a converting unit, andwherein:

-   -   the generating unit accepts generated inputs, x₁, x₂, . . . ,        x_(n)εA, whose length is “n” in total, and generates generated        outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n” in total        using recurrence formulas

y ₁ =F ₁(x ₁ ,a), and

y _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1);

-   -   the key accepting unit accepts key inputs, k₁, k₂, . . . ,        k_(n)εA, whose length is “n” in total, and gives the accepted        key inputs as generated inputs to the generating unit;    -   the repetition controller gives the generated outputs from the        generating unit as generated inputs to the generating unit, for        an “m” (m≧0) number of times, and sets one of the generated        outputs to be given at end as a random number string, r₁, r₂, .        . . , r_(n)εA, whose length is “n” in total;    -   the data accepting unit accepts data inputs, d₁, d₂, . . . ,        d_(n)εA, whose length is “n” in total; and    -   the converting unit converts data for any integers “i” in a        range between 1 and “n” using a formula

e_(i)=d_(i)⋆r_(i),and

-   -   -   outputs data outputs, e₁, e₂, . . . , e_(n)εA, whose length            is “n” in total.

In order to accomplish the above object, according to the second aspectof the present invention, there is provided a converter using:

an “n” (n≧1) number of conversion functions, F_(i): A×A→A (1≦i≦n), withrespect to a domain A;

a binary arithmetic operation, ⋆: A×A→A, and its reverse binaryarithmetic operation, ⊚: A×A→A, wherein,

-   -   for arbitrary xεA, yεA, conditions of

(x⋆y)⊚y=x, and

(x⊚y)⋆y=x

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converter comprising a generating unit, a key accepting unit, arepetition controller, a data accepting unit, and a converting unit, andwherein:

-   -   the generating unit accepts generated inputs, x₁, x₂, . . . ,        x_(n)εA, whose length is “n” in total, and generates generated        outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n” in total        using recurrence formulas,

y ₁ =F ₁(x ₁ ,a), and

y _(i−1) =F _(i+1)(x _(i+1) ,x ₁)(1≦i≦n−1);

-   -   the key accepting unit accepts key inputs, k₁, k₂, . . . ,        k_(n)εA whose length is “n” in total, and gives the accepted key        inputs as generated inputs to the generating unit;    -   the repetition controller gives the generated outputs from the        generating unit as generated inputs to the generating unit, for        an “m” (m≧0) number of times, and sets one of the generated        outputs to be given at end as a random number string, r₁, r₂, .        . . , r_(n)εA, whose length is “n” in total;    -   the data accepting unit accepts data inputs, d₁, d₂, . . . ,        d_(n)εA, whose length is “n” in total; and    -   the converting unit converts data for any integers “i” in a        range between 1 and “n” using a formula

e_(i)=d_(i)⋆r_(i), and

-   -   -   outputs data outputs, e₁, e₂, . . . , e_(n)εA, whose length            is “n” in total.

In the above converter, each of the binary arithmetic operations ⊚ and ⋆may be exclusive OR.

In the above converter,

at least one of the conversion functions F_(i) may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter b (1≦b≦M^(s)),

F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b), and

F _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)),

in cases where:

-   -   “ceil (•)” represents that decimals should be rounded off to a        next whole number in “M” number system; and    -   “floor (•)” represents that decimals should be cut off in “M”        number system.

In the converter,

at least one of the conversion functions F_(i) may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter, b (1≦b≦M^(s)),

F _(i)(y,b)=x ₁(q<x ₁);

F _(i)(y,b)=x ₂(x ₁ ≦q),

where

x ₁=floor(M ^(−s) by);

x ₂ =ceil((M ^(−s) b−1)y+M ^(s));

q=b(x ₂ −M ^(s))/(b−M ^(s)),

in cases where:

-   -   “ceil (•)” represents that decimals should be rounded off to a        next whole number in “M” number system; and    -   “floor (•)” represents that decimals should be cut off in “M”        number system.

In order to accomplish the above object, according to the third aspectof the present invention, there is provided an encryption/decryptionsystem including the above-described converter as an encrypting unit andanother converter having a same structure of a structure of theconverter as a decrypting unit, and wherein:

“F_(i)”, ⊚, and “a”, are commonly used by the encrypting unit and thedecrypting unit;

a condition, x⋆y=x⊚y, is satisfied for an arbitrary xεA and yεA;

the encrypting unit and the decrypting unit commonly accepts key inputs,k₁, k₂, . . . , k_(n);

the encrypting unit accepts original data whose length is “n”, as a datainput, and outputs a data output whose length is “n” as encrypted data;and

the decrypting unit accepts the encrypted data whose length is “n”, as adata input, and outputs a data output whose length is “n” as decrypteddata.

In order to accomplish the above object, according to the fourth aspectof the present invention, there is provided a converter using:

an “n” (n≧1) number of conversion functions F_(i): A×A→A (1≦i≦n) andtheir reverse conversion functions G₁: A×A→A, with respect to a domainA, wherein, for arbitrary xεA and yεA, conditions of

F _(i)(G _(i)(x,y),y)=x, and

G _(i)(F _(i)(x,y),y)=x,

-   -   are satisfied;

a binary arithmetic operation, ⋆: A^(n)→A^(n), and its reverse binaryarithmetic operation, ⊚: A^(n)→A^(n), wherein, for arbitrary zεA^(n),conditions of

⋆(⊚z)=z, and

⊚(⋆z)=z

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converter comprising a generating unit, a data accepting unit, arepetition controller, and a converting unit, and wherein:

-   -   the generating unit accepts generated inputs, x₁, x₂, . . . ,        x_(n)εA, whose length is “n” in total, and generates generated        outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n” in total        using recurrence formulas

y ₁ =F ₁(x ₁ ,a), and

y _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1);

-   -   the data accepting unit accepts data inputs, k₁, k₂, . . . ,        k_(n)εA, whose length is “n” in total, and gives the accepted        data inputs as generated inputs to the generating unit;    -   the repetition controller gives the generated outputs from the        generating unit as generated inputs to the generating unit, for        an “m” (m≧0) number of times, and sets one of the generated        outputs to be given at end as a random number string, r₁, r₂, .        . . , r_(n)εA, whose length is “n” in total; and    -   the converting unit applies a single-term arithmetic operation,        ⋆, to the random number string, r₁, r₂, . . . , r_(n)εA, to        perform its data conversion, that is,

(e ₁ ,e ₂ , . . . , e _(n))=(r ₁ ,r ₂ , . . . , r _(n)), and

-   -   -   outputs data outputs, e₁, e₂, . . . , e_(n), whose length is            “n” in total.

In order to accomplish the above object, according to the fifth aspectof the present invention, there is provided a converter using:

an “n” (n≧1) number of conversion functions, F_(i): A×A→A (1≦i≦n), andtheir reverse conversion functions, G_(i): A×A→A, with respect to adomain A, wherein, for arbitrary xεA and yεA, conditions of

F _(i)(G _(i)(x,y),y)=x, and

G _(i)(F _(i)(x,y),y)=x,

-   -   are satisfied;

a binary arithmetic operation, ⋆: A^(n)→A^(n), and its reverse binaryarithmetic operation, ⊚: A^(n)→A^(n), wherein, for arbitrary zεA^(n),conditions of

⋆(⊚z)=z, and

⊚(⋆z)=z,

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converter comprising a generating unit, a data accepting unit, aconverting unit, and a repetition controller, and wherein:

-   -   the generating unit accepts generated inputs, x₁, x₂, . . . ,        x_(n)εA, whose length is “n” in total, and generates generated        outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n” in total        using recurrence formulas,

y ₁ =G ₁(x ₁ ,a), and

y _(i+1) =G _(i+1)(x _(i+1) ,x _(i))(1≦i≦n−1);

-   -   the data accepting unit accepts data inputs, h₁, h₂, . . . ,        h_(n)εA, whose length is “n” in total;    -   the converting unit applies a single-term arithmetic operation,        ⋆, to the data inputs, h₁, h₂, . . . , h_(n), to perform its        data conversion, that is,

(v ₁ ,v ₂ , . . . , v _(n))=⋆(h ₁ ,h ₂ , . . . , h _(n)), and

-   -   -   gives results of the data conversion, v₁, v₂, . . . , v_(n),            to the generating unit; and

    -   the repetition controller gives the generated outputs from the        generating unit as generated inputs to the generating unit, for        an “m” (m≧0) number of times, and sets one of the generated        outputs to be given at end as data outputs, s₁, s₂, . . . ,        s_(n)εA, whose length is “n” in total.

In the above converter,

in cases where “A” represents a “t”-number bit space, and “zεA^(n)”corresponds to a bit string having “tn” bits in length, in thesingle-term arithmetic operation ⊚, bits in the bit string may beshifted by a predetermined number of bits in a predetermined direction,and its resultant bit string may be set to correspond to A^(n), therebyobtaining a result of the single-term arithmetic operation ∘.

In the converter,

at least one of the conversion functions, F_(i), may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter b (1≦b≦M^(s)),

F _(i)(x,b)ceil(xM ^(s) /b)(1≦x≦b), and

F _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)),

in cases where:

-   -   “ceil (•)” represents that decimals should be rounded off to a        next whole number in “M” number system; and    -   “floor (•)” represents that decimals should be cut off in “M”        number system.

In the converter,

at least one of the conversion functions, F_(i), may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter, b (1≦b≦M^(s)),

F _(i)(y,b)=x ₁(q<x ₁);

F _(i)(y,b)=x ₂(x≦q),

where

x ₁=floor(M ^(−s) by);

x ₂=ceil((M ^(−s) b−1)y+M ^(s));

q=b(x ₂ −M ^(s))/(b−M ^(s)),

-   -   in cases where:        -   “ceil (•)” represents that decimals should be rounded off to            a next whole number in “M” number system; and        -   “floor (•)” represents that decimals should be cut off in            “M” number system.

In order to accomplish the above object, according to the sixth aspectof the present invention, there is provided as encryption/decryptionsystem including the above-described former converter as an encryptingunit and the above-described latter converter as a decrypting unit, andwherein:

“F_(i)”, “G_(i)”, “⋆”, “⊚”, and “a”, are commonly used by the encryptingunit and the decrypting unit;

the encrypting unit accepts original data as data inputs, k₁, k₂, . . ., k_(n), whose length is “n” in total, and outputs data outputs, e₁, e₂,. . . , e_(n), whose length is “n” in total as encrypted data; and

the decrypting unit accepts the encrypted data whose length is “n” intotal, as data inputs, h₁, h₂, . . . , h_(n), and outputs data outputs,s₁, s₂, . . . , s_(n), whose length is “n” in total as decrypted data.

In order to accomplish the above object, according to the seventh aspectof the present invention, there is provided an encryption/decryptionsystem including the above-described former converter as an encryptingunit and the above-described latter converter as a decrypting unit, andwherein:

“F_(i)”, “G_(i)”, “⋆”, “⊚”, and “a” are commonly used by the encryptingunit and the decrypting unit;

the encrypting unit accepts original data as data inputs, h₁, h₂, . . ., h_(n), whose length is “n” in total, and outputs data outputs, s₁, s₂,. . . , s_(n), whose length is “n” in total as encrypted data; and

the decrypting unit accepts the encrypted data whose length is “n” intotal, as data inputs, k₁, k₂, . . . , k_(n), and outputs data outputs,e₁, e₂, . . . , e_(n), whose length is “n” in total as decrypted data.

In order to accomplish the above object, according to the eighth aspectof the present invention, there is provided a multi-stage convertercomprising:

a “u” number of above-described latter converters (a “j”-th converter iscalled a converter M_(j) (1≦j≦u)); and

a multi-stage key-input accepting unit which accepts parameter inputsa₁, a₂, . . . , a_(n)εA, and sets a “j”-th parameter input, a_(j),included in the accepted parameter inputs, as a predetermined parameter“a” of the converter M_(j), and wherein

a converter M₁ included in the “u” number of converters acceptsmulti-stage conversion inputs, k₁, k₂, . . . , k_(n), whose length is“n” in total, as data inputs,

data outputs, which are output by a converter M_(i) (1≦i≦u−1) includedin the “u” number of converters, are given to a converter M_(i+1)included in the “u” number of converters, as data inputs, and

a converter M_(u) included in the “u” number of converters outputs dataoutputs, e₁, e₂, . . . , e_(n), whose length is “n” in total, asmulti-stage conversion outputs.

In order to accomplish the above object, according to the ninth aspectof the present invention, there is provided a multi-stage convertercomprising:

a “u” number of above-described latter converters (a “j”-th converter iscalled a converter M_(j) (1≦j≦u)) according to claim 7; and

a multi-stage key-input accepting unit which accepts parameter inputsa₁, a₂, . . . , a_(u)εA, and sets a “j”-th parameter input, a_(j),included in the accepted parameter inputs, as a predetermined parameter“a” of the converter M_(j), and wherein

a converter M_(u) included in the “u” number of converters acceptsmulti-stage conversion inputs, h₁, h₂, . . . , h_(n), whose length is“n” in total, as data inputs,

data outputs, which are output by a converter M_(i+1) (1≦i≦u−1) includedin the “u” number of converters, are given to a converter M_(i) includedin the “u” number of converters, as data inputs, and

a converter M₁ included in the “u” number of converters outputs dataoutputs, s₁, s₂, . . . , s_(n), whose length is “n” in total, asmulti-stage conversion outputs.

In order to accomplish the above object, according to the tenth aspectof the present invention, there is provided an encryption/decryptionsystem including the above-described former multi-stage converter as anencrypting unit and the above-described latter multi-stage converter asa decrypting unit, and wherein:

“F_(i)”, “G_(i)”, “⋆”, and “⊚”, are commonly used by the encrypting unitand the decrypting unit;

parameter inputs, a₁, a₂, . . . , a_(u), are commonly accepted by theencrypting unit and the decrypting unit;

the encrypting unit accepts original data as multi-stage conversioninputs, k₁, k₂, . . . , k_(n), whose length is “n” in total, and outputsmulti-stage conversion outputs, e₁, e₂, . . . , e_(n), whose length is“n” in total as encrypted data; and

the decrypting unit accepts the encrypted data whose length is “n” intotal, as multi-stage conversion inputs, h₁, h₂, . . . , h_(n), andoutputs data outputs, s₁, s₂, . . . , s_(n), whose length is “n” intotal as decrypted data.

In order to accomplish the above object, according to the eleventhaspect of the present invention, there is provided anencryption/decryption system including the above-described lattermulti-stage converter as an encrypting unit and the above-describedformer multi-stage converter as a decrypting unit, and wherein:

“F_(i)”, “G_(i)”, “⋆”, and “⊚”, are commonly used by the encrypting unitand the decrypting unit;

parameter inputs, a₁, a₂, . . . , a_(u), are commonly accepted by theencrypting unit and the decrypting unit;

the encrypting unit accepts original data as multi-stage conversioninputs, h₁, h₂, . . . , h_(n), whose length is “n” in total, and outputsmulti-stage conversion outputs, s₁, s₂, . . . , s_(n), whose length is“n” in total as encrypted data; and

the decrypting unit accepts the encrypted data whose length is “n” intotal, as multi-stage conversion inputs, k₁, k₂, . . . , k_(n), andoutputs data outputs, e₁, e₂, . . . , e₁, whose length is “n” in totalas decrypted data.

In order to accomplish the above object, according to the twelfth aspectof the present invention, there is provided a converting method using:

an “n” (n≧1) number of conversion functions, F_(i): A×A→A (1≦i≦n), withrespect to a domain A;

a binary arithmetic operation, ⋆: A×A→A, and its reverse binaryarithmetic operation, ⊚: A×A→A, wherein,

-   -   for arbitrary xεA, yεA, conditions of

(x⋆y)⊚y=x, and

(x⊚y)⋆y=x

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converting method comprising a generating step, a key acceptingstep, a repetition controlling step, a data accepting step, and aconverting step, and wherein:

-   -   the generating step includes accepting generated inputs, x₁, x₂,        . . . , x_(n)εA, whose length is “n” in total, and generating        generated outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n”        in total using recurrence formulas,

y ₁ =F ₁(x ₁ ,a), and

y _(i+1) =F _(i+1)(x _(i+1) y _(i))(1≦i≦n−1);

-   -   the key accepting step includes accepting key inputs, k₁, k₂, .        . . , k_(n)εA, whose length is “n” in total, and giving the        accepted key inputs as generated inputs to the generating step;    -   the repetition controlling step includes giving the generated        outputs from the generating step as generated inputs to the        generating step, for an “m” (m≧0) number of times, and setting        one of the generated outputs to be given at end as a random        number string, r₁, r₂, . . . , r_(n)εA, whose length is “n” in        total;    -   the data accepting step includes accepting data inputs, d₁, d₂,        . . . , d_(n)εA, whose length is “n” in total; and    -   the converting step includes converting data for any integers        “i” in a range between 1 and “n” using a formula,

e_(i)=d_(i)⋆r_(i), and

-   -   -   outputting data outputs, e₁, e₂, . . . , e_(n)εA, whose            length is “n” in total.

In order to accomplish the above object, according to the thirteenthaspect of the present invention, there is provided a converting methodusing:

an “n” (n≧1) number of conversion functions, F_(i): A×A→A (1≦i≦n), withrespect to a domain A;

a binary arithmetic operation, ⋆: A×A→A, and its reverse binaryarithmetic operation, ⊚: A×A→A, wherein,

-   -   for arbitrary xεA, yεA, conditions of

(x⋆y)⊚y=x, and

(x⊚y)⋆y=x

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converting method comprising a generating step, a key acceptingstep, a repetition controlling step, a data accepting step, and aconverting step, and wherein:

-   -   the generating step includes accepting generated inputs, x₁, x₂,        . . . , x_(n)εA, whose length is “n” in total, and generating        generated outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n”        in total using recurrence formulas,

y ₁ =F ₁(x ₁ ,a), and

y _(i+1) =F _(i+1)(x _(i+1) ,x _(i))(1≦i≦n−1);

-   -   the key accepting step includes accepting key inputs, k₁, k₂, .        . . , k_(n)εA whose length is “n” in total, and giving the        accepted key inputs as generated inputs to the generating step;    -   the repetition controlling step includes giving the generated        outputs from the generating step as generated inputs to the        generating step, for an “m” (m≧0) number of times, and setting        one of the generated outputs to be given at end as a random        number string, r₁, r₂, . . . , r_(n)εA, whose length is “n” in        total;    -   the data accepting step includes accepting data inputs, d₁, d₂,        . . . , d_(n)εA, whose length is “n” in total; and    -   the converting step includes converting data for any integers        “i” in a range between 1 and “n” using a formula

e_(i)=d_(i)⋆r_(i), and

-   -   -   outputting data outputs, e₁, e₂, . . . , e_(n)εA, whose            length is “n” in total.

Each of the binary arithmetic operations ⊚ and ⋆ may be exclusive OR.

In the converting method,

at least one of the conversion functions F_(i) may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter b (1≦b≦M^(s)),

F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b), and

F _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)),

in cases where:

-   -   “ceil (•)” represents that decimals should be rounded off to a        next whole number in “M” number system; and    -   “floor (•)” represents that decimals should be cut off in “M”        number system.

In the converting method,

at least one of the conversion functions F_(i) may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter, b (1≦b≦M^(s)),

F _(i)(y,b)=x ₁(q<x ₁);

F _(i)(y,b)=x ₂(x ₁ ≦q),

where

x ₁ =floor(M ^(−s) by);

x ₂ =ceil((M ^(−s) b−1)y+M ^(s));

q=b(x ₂ −M ^(s))/(b−M ^(s)),

in cases where:

-   -   “ceil (•)” represents that decimals should be rounded off to a        next whole number in “M” number system; and    -   “floor (•)” represents that decimals should be cut off in “M”        number system.

In order to accomplish the above object, according to the fourteenthaspect of the present invention, there is provided a converting methodusing:

an “n” (n≧1) number of conversion functions, F_(i): A×A→A, (1≦i≦n) andtheir reverse conversion functions, G_(i): A×A→A, with respect to adomain A, wherein, for arbitrary xεA and yεA, conditions of

F _(i)(G _(i)(x,y),y)=x, and

G _(i)(F _(i)(x,y),y)=x,

-   -   are satisfied;

a binary arithmetic operation, ⋆: A^(n)→A^(n), and its reverse binaryarithmetic operation, ⊚: A^(n)→A^(n), wherein, for arbitrary zεA^(n),conditions of

⋆(⊚z)=z, and

⊚(⋆z)=z

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converting method comprising a generating step, a data acceptingstep, a repetition controlling step, and a converting step, and wherein:

-   -   the generating step includes accepting generated inputs, x₁, x₂,        . . . , x_(n)εA, whose length is “n” in total, and generating        generated outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n”        in total using recurrence formulas,

y ₁ =F ₁(x ₁ ,a), and

y _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1);

-   -   the data accepting step includes accepting data inputs, k₁, k₂,        . . . , k_(n)εA, whose length is “n” in total, and giving the        accepted data inputs as generated inputs to the generating step;    -   the repetition controlling step includes giving the generated        outputs from the generating step as generated inputs to the        generating step, for an “m” (m≧0) number of times, and setting        one of the generated outputs to be given at end as a random        number string, r₁, r₂, . . . , r_(n)εA, whose length is “n” in        total; and

the converting step includes applying a single-term arithmeticoperation, ⋆, to the random number string, r₁, r₂, . . . , r_(n)εA, toperform its data conversion, that is,

(e ₁ ,e ₂ , . . . , e _(n))=⋆(r ₁ ,r ₂ , . . . , r _(n)), and

-   -   outputting data outputs, e₁, e₂, . . . , e_(n), whose length is        “n” in total.

In order to accomplish the above object, according to the fifteenthaspect of the present invention, there is provided a converting methodusing:

an “n” (n≧1) number of conversion functions, F_(i): A×A→A (1≦i≦n), andtheir reverse conversion functions, G_(i): A×A→A, with respect to adomain A, wherein, for arbitrary xεA and yεA, conditions of

F _(i)(G _(i)(x,y),y)=x, and

G _(i)(F _(i)(x,y),y)=x,

-   -   are satisfied;

a binary arithmetic operation, ⋆: A^(n)→A^(n), and its reverse binaryarithmetic operation, ⊚: A^(n)→A^(n), wherein, for arbitrary zεA^(n),conditions of

⋆(⊚z)=z, and

⊚(⋆z)=z

-   -   are satisfied; and

a predetermined parameter, aεA, and

the converting method comprising a generating step, a data acceptingstep, a converting step, and a repetition controlling step, and wherein:

-   -   the generating step includes accepting generated inputs, x₁, x₂,        . . . , x₁εA, whose length is “n” in total, and generating        generated outputs, y₁, y₂, . . . , y_(n)εA, whose length is “n”        in total using recurrence formulas,

y ₁ =G ₁(x ₁ ,a), and

Y _(i+1) =G _(i+1)(x _(i+1) ,x _(i))(1≦i≦n−1);

-   -   the data accepting step includes accepting data inputs, h₁, h₂,        . . . , h_(n)εA, whose length is “n” in total;    -   the converting step includes applying a single-term arithmetic        operation, ⋆, to the data inputs, h₁, h₂, . . . , h_(n), to        perform its data conversion, that is,

(v ₁ ,v ₂ , . . . , v _(n))=(h ₁ ,h ₂ , . . . , h _(n)), and

-   -   giving results of the data conversion, v₁, v₂, . . . , v_(n), to        the generating step; and

the repetition controlling step includes giving the generated outputsfrom the generating step as generated inputs to the generating step, foran “m” (m≧0) number of times, and setting one of the generated outputsto be given at end as data outputs, s₁, s₂, . . . , s_(n)εA, whoselength is “n” in total.

In the above-described converting method, in cases where “A” representsa “t”-number bit space, and “zεA^(n)” corresponds to a bit string having“tn” bits in length, in the single-term arithmetic operation ⊚, bits inthe bit string may be shifted by a predetermined number of bits in apredetermined direction, and its resultant bit string may be set tocorrespond to A^(n), thereby obtaining a result of the single-termarithmetic operation ⊚.

In the converting method,

at least one of the conversion functions F_(i) may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter b (1≦b≦M^(s)),

F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b), and

F _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)).

in cases where:

-   -   “ceil (•)” represents that decimals should be rounded off to a        next whole number in “M” number system; and    -   “floor (•)” represents that decimals should be cut off in “M”        number system.

In the converting method,

at least one of the conversion functions F_(i) may define positiveintegers M, s, and satisfy following conditions, for an arbitraryinteger parameter, b (1≦b≦M^(s)),

F _(i)(y,b)=x ₁(q<x ₁);

F _(i)(y,b)=x ₂(x ₁ ≦q),

where

x ₁=floor(M ^(−s) by);

x ₂=ceil((M ^(−s) b−1)y+M ^(s));

q=b(x ₂ −M ^(s))/(b−M ^(s)),

in cases where:

-   -   “ceil (•)” represents that decimals should be rounded off to a        next whole number in “M” number system; and    -   “floor (•)” represents that decimals should be cut off in “M”        number system.

In order to accomplish the above object, according to the sixteenthaspect of the present invention, there is provided a multi-stageconverting method comprising:

a “u” number of converting steps (a “j”-th converting step is called aconverting step M_(j) (1≦j≦u)) of using the converting method accordingto claim 23; and

a multi-stage key-input accepting step of accepting parameter inputs a₁,a₂, . . . , a_(u)εA whose length is “n” in total, and setting a “j”-thparameter input, a_(j), included in the accepted parameter inputs, as apredetermined parameter “a” of the converting step M_(j), and wherein

a converting step M₁ included in the “u” number of converting stepsincludes accepting multi-stage conversion inputs, k₁, k₂, . . . , k_(n),whose length is “n” in total, as data inputs,

data outputs, which are output at a converting step M_(i) (1≦i≦u−1)included in the “u” number of converting steps, are given to aconverting step M_(i+1) included in the “u” number of converting steps,as data inputs, and

a converting step M_(u) included in the “u” number of converting stepsincludes outputting data outputs, e₁, e₂, . . . , e_(n), whose length is“n” in total, as multi-stage conversion outputs.

In order to accomplish the above object, according to the seventeenthaspect of the present invention, there is provided a multi-stageconverting method comprising:

a “u” number of converting steps (a “j”-th converting step is called aconverting step M_(j) (1≦j≦u)) of using the converting method accordingto claim 24; and

a multi-stage key-input accepting step of accepting parameter inputs a₁,a₂, . . . , a_(u)εA whose length is “n” in total, and setting a “j”-thparameter input, a_(j), included in the accepted parameter inputs, as apredetermined parameter “a” of the converting step M_(j), and wherein

a converting step M_(u) included in the “u” number of converting stepsincludes accepting multi-stage conversion inputs, h₁, h₂, . . . , h_(n),whose length is “n” in total, as data inputs,

data outputs, which are output at a converting step M_(i+1) (1≦i≦u−1)included in the “u” number of converting steps, are given to aconverting step M_(i) included in the “u” number of converting steps, asdata inputs, and

a converting step M₁ included in the “u” number of converting stepsincludes outputting data outputs, s₁, s₂, . . . s_(n), whose length is“n” in total, as multi-stage conversion outputs.

In order to accomplish the above object, according to the eighteenthaspect of the present invention, there is provided a program forcontrolling a computer to serve as any of the above-described convertersor any of the above-described multi-stage converters, or a program forcontrolling a computer to execute any of the above-described convertingmethods or any of the above-described multi-stage converting methods.

In order to accomplish the above object, according to the nineteenthaspect of the present invention, there is provided an informationrecording medium recording any of the programs.

As the above-described information recording medium, there may beemployed a compact disk, a flexible disk, a hard disk, a magneto-opticaldisk, a digital video disk, a magnetic tape, and a semiconductor memory.

Separately from the computer to be executing the program, the program ofthe present invention may be distributed or sold through a computercommunication network. In addition, separately from the computer to beexecuting the program, the information recording medium of the presentinvention may be distributed or sold through general businesstransactions, etc.

BRIEF DESCRIPTION OF THE DRAWINGS

The object and other objects and advantages of the present inventionwill become more apparent upon reading of the following detaileddescription and the accompanying drawings in which:

FIG. 1 is an exemplary diagram showing the schematic structure of aconverter according to the first embodiment of the present invention;

FIG. 2 is a flowchart showing procedures of a conversion process whichis carried out by a serial computer serving as the converter;

FIG. 3 is an exemplary diagram showing the schematic structure of aconverter according to the second embodiment of the present invention;

FIG. 4 is a flowchart showing procedures of a conversion process whichis carried out by a serial computer serving as the converter of FIG. 3;

FIG. 5 is an exemplary diagram showing the schematic structure of anencryption/decryption system including the converters respectively as anencrypting unit and a decrypting unit;

FIG. 6 is an exemplary diagram showing the schematic structure of anencryption/decryption system including the converters respectively as anencrypting unit and a decrypting unit;

FIG. 7 is an exemplary diagram showing the schematic structure of aconverter according to the fourth embodiment of the present invention;

FIG. 8 is a flowchart showing procedures of a conversion process whichis carried out by a serial computer serving as the converter of FIG. 7;

FIG. 9 is an exemplary diagram showing the schematic structure of aconverter according to the fifth embodiment of the present invention;

FIG. 10 is a flowchart showing procedures of a conversion process whichis carried out by a serial computer serving as the converter of FIG. 9;

FIG. 11 is an exemplary diagram showing an encryption/decryption systemincluding the converter as an encrypting unit and the converter as adecrypting unit;

FIG. 12 is an exemplary diagram showing an encryption/decryption systemincluding the converter as an encrypting unit and the converter as adecryption unit;

FIG. 13 is an exemplary diagram showing the schematic structure of amulti-stage converter according to the seventh embodiment of the presentinvention;

FIG. 14 is an exemplary diagram showing the schematic structure of amulti-stage converter according to the eighth embodiment of the presentinvention;

FIG. 15 is an exemplary diagram showing the schematic structure of anencryption/decryption system, according to the ninth embodiment of thepresent invention, including the multi-stage converters which are in apair relationship with each other;

FIG. 16 is an exemplary diagram showing the schematic structure of anencryption/decryption system, according to the tenth embodiment of thepresent invention, including the multi-stage converters which are in apair relationship with each other; and

FIG. 17 is a distribution diagram showing a distribution of datagenerated according to the technique of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Preferred embodiments for practicing the present invention will now bedescribed. Embodiments, as will be explained later, are to illustratethe present invention, not to limit the scope of the present invention.For those skilled in the art, the present invention may be applicable toembodiments including replaced elements equivalent to each or entireelements of the present invention, and such embodiments are, therefore,within the scope of the present invention.

In the explanations below, a converter which can be adopted for anencryption system using a vector-stream private (secret) key will bedescribed in each of the first and second embodiments of the presentinvention, and an encryption/decryption system using either encryptionsystem of the first and second embodiments will be described in thethird embodiment of the present invention.

In the preferred embodiments of the present invention, with respect to adomain A, there are an “n” (n≧1) number of conversion function(s) F_(i):A×A→A (1≦i≦n), a binary arithmetic operation ⋆: A×A→A, and its reversebinary arithmetic operation ⊚: A×A→A. In this case, for arbitrary xεAand yεA, the conditions of: (x⋆y) ⊚y=x; and (x⊚y) ⋆y=x should besatisfied.

As such binary arithmetic operations ⊚ and ⋆, exclusive OR will beemployed in the following embodiments.

In the following explanations, “ceil (•)” represents that decimalsshould be rounded off to the next whole number in “M” number system, and“floor (•)” represents that decimals should be cut off in “M” numbersystem.

In the following embodiments, at least one of conversion functions F_(i)defined by positive integers M, s, and should satisfy the followingconditions of:

F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b);

F _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b<x≦M ^(s)),

for an arbitrary integer parameter b (1≦b≦M^(s)). This conversionfunction corresponds to Masuda-Aihara mapping with a parameter (IEICETrans. on Communication, 1999, July, Vol. J82-A, No. 7, pp. 1042-1046).This mapping is called also a skew tent mapping.

In the following embodiments, instead of the above-described conversionfunctions F_(i), there can be employed a function (reverse mapping ofthe above-described Masuda-Aihara mapping with a parameter) which isdefined by positive integers M, s, and satisfies, for an arbitraryinteger parameter b (1≦b≦M^(s)), the following conditions of:

F _(i)(y,b)=x ₁(q<x ₁);

F _(i)(y,b)=x ₂(x ₁ ≦q),

where

x ₁=floor(M ^(−s) by);

x ₂ =ceil((M ^(−s) b−1)y+M ^(s));

q=b(x ₂ −M ^(s))/(b−M ^(s)).

And in the following embodiments, instead of the above-describedconversion functions F_(i), there can be employed a function which isdefined by positive integers M, s, is a second degree polynomial in xover module M^(s), and satisfies, for an arbitrary integer parameter b(1≦b≦M^(s)) and a predefined function g of b, the following conditionsof:

F _(i)(x,b)=2x(x+g(b))mod M ^(s).

First Embodiment

FIG. 1 is an exemplary diagram showing the schematic structure of aconverter according to the first embodiment of the present invention.

A converter 101 uses a predetermined parameter aεA. The converter 101includes a generating unit 102, a key accepting unit 103, a repetitioncontroller 104, a data accepting unit 105, a converting unit 106.

The generating unit 102 receives generated inputs, x₁, x₂, . . . ,x_(n)εA, whose length is “n” in total, and

generates generated outputs, y₁, y₂, . . . , y_(n)εA, whose length is“n” in total, using the following recurrence formulas:

y ₁ =F ₁(x ₁ ,a);

y _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1).

The key accepting unit 103 accepts key inputs, k₁, k₂, . . . , k_(n)εAwhose length is “n”, and gives the generating unit 102 the accepted keyinputs.

The repetition controller 104 gives back the generating unit 102 thegenerated outputs from the generating unit 102 as generated inputs,repeatedly for an “m” (m≧0) number of times. In this case, the generatedoutputs to be given at the end is a random number string, r₁, r₂, . . ., r_(n)εA, whose length is “n” in total.

The data accepting unit 105 accepts data inputs, d₁, d₂, . . . ,d_(n)εA, whose length is “n” in total.

The converting unit 106 performs data conversion for any integer(s) “i”in a range between 1 and “n”, using the formula

e_(i)=d_(i)⋆r_(i),

so as to output data outputs, e₁, e₂, . . . , e_(n)εA, whose length is“n” in total.

This calculation (data conversion) can be executed at high speed by aparallel computer having a pipeline process function. However, in thefollowing explanations, the above calculation is to be executed by agenerally-used serial computer.

FIG. 2 is a flowchart for explaining a conversion process which iscarried out by a serial computer serving as the converter 101.

The converter 101 accepts key input variables, k₁, k₂, . . . , k_(n)εA(Step S201).

The converter 101 substitutes the accepted variables respectively forvariables x₁, x₂, . . . , x_(n)εA (Step S202).

After this, the converter 101 substitutes a value “m” for a countervariable “c” (Step S203).

Further, the converter 101 calculates variables, y₁, y₂, . . . , y_(n)εA(Step S204), using the following recurrence formulas:

y ₁ =F ₁(x ₁ ,a);

y _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1).

The converter 101 checks whether the counter variable “c” is 0 (StepS205). In the case where it is determined that the counter variable “c”is not 0 (Step S205; No), the converter 101 substitutes the variables,y₁, y₂, . . . , y_(n) for the variables x₁, x₂, . . . , x_(n) (StepS206). After this, the converter 101 decrements the counter variable “c”by 1 (Step S207), and the flow returns to the procedure of the stepS204.

In the case where it is determined that the counter variable “c” is 0(Step S205; Yes), the converter 101 substitutes the variables, y₁, y₂, .. . , y_(n) for variables r₁, r₂, . . . , r_(n)εA (Step S208).

The converter 101 accepts target data inputs, d₁, d₂, . . . , d_(n)εA tobe encrypted (Step S209).

The converter 101 performs data conversion for any integer(s) “i” in arange between 1 and “n”, using the formula

e_(i)=d_(i)⋆r_(i)  (Step S210).

Finally, the converter 101 outputs variables, e₁, e₂, . . . , e_(n)(Step S211).

By the above-described processes, the conversion process to be adoptedin the encryption/decryption system of the present invention will berealized.

Second Embodiment

FIG. 3 is an exemplary diagram showing the schematic structure of aconverter according to the second embodiment of the present invention.The converter according to this embodiment will now specifically beexplained with reference to FIG. 3.

A converter 301 has the structure which is substantially the same as thestructure of the converter 101. The converter 301 has a generating unit302 corresponding to the generating unit 102, a key accepting unit 303corresponding to the key accepting unit 103, a repetition controller 304corresponding to the repetition controller 104, a data accepting unit305 corresponding to the data accepting unit 105, and a converting unit306 corresponding to the converting unit 106.

The generating unit 302 uses recurrence formulas which are differentfrom the recurrence formulas used by the generating unit 102.Specifically, the generating unit 302 uses recurrence formulas:

y ₁ =F ₁(x ₁ ,a);

y _(i+1) =F _(i+1)(x _(i+1) ,x _(i))(1≦i≦n−1).

FIG. 4 is a flowchart for explaining a conversion process which iscarried out by a serial computer serving as the converter 301. Theprocedures of the conversion process which are performed by theconverter 301 are substantially the same as those of conversion processperformed by the converter 101, and the procedures of the steps S401 toS411 to be executed by the converter 301 respectively correspond to theprocedures of the steps S201 to S211 to be executed by the converter101.

The recurrence formulas used in the step S404 differ from the recurrenceformulas used in the step S204. That is, in the step S404, the converter301 uses the recurrence formulas:

y ₁ =F ₁(x ₁ ,a);

y _(i+1) =F _(i+1)(x _(i+1) ,x _(i))(1≦i≦n−1).

Third Embodiment

An encryption/decryption system according to the third embodiment of thepresent invention includes either the converter 101 or the converter 301as an encrypting unit, and further includes the same as a decryptingunit.

FIG. 5 is an exemplary diagram showing the schematic structure of theencryption/decryption system including two converters 101 serving as theencrypting unit and a decrypting unit.

An encryption/decryption system 501 includes an encrypting unit 502 anda decrypting unit 503. Each of the encrypting unit 502 and thedecrypting unit 503 includes the converter 101.

The encrypting unit 502 and the decrypting unit 503 use the same “F_(i)”and “a”. In this embodiment, the symbols ⊚ and ⋆ express the function ofexclusive OR, so that a condition of x⋆y=x⊚y should be satisfied, forarbitrary xεA, yεA.

Each of the encrypting unit 502 and the decrypting unit 503 accepts thecommon key inputs, k₁, k₂, . . . , k_(n).

The encrypting unit 502 accepts original data whose length is “n” intotal, as data inputs, and outputs data outputs whose length is “n” intotal, as encrypted data.

The decrypting unit 503 accepts the encoded data, whose length is “n” inlength, as data inputs, and outputs data outputs, whose length is “n” intotal, as decoded data.

In this manner, a vector-stream private key encryption system can thusbe realized.

FIG. 6 is an exemplary diagram showing the schematic structure of anencryption/decryption system including two converters 301 which serve asan encrypting unit and a decrypting unit. In this embodiment also, theencryption/decryption system 501 includes the encrypting unit 502 andthe decrypting unit 503. Except that the each of the encrypting unit 502and the decrypting unit 503 includes the converter 301, theencryption/decryption system 501 has the same structure as that of FIG.5.

According to this embodiment also, a vector-stream private keyencryption system can be realized.

In the explanations below, a converter which can be adopted for avector-stream private key encryption system will be described in each ofthe fourth and fifth embodiments of the present invention, and anencryption/decryption system using either encryption system of thefourth and fifth embodiments will be described in the sixth embodimentof the present invention.

In the explanations below, there are employed an “n” (1≦n) number ofconversion functions F_(i): A×A→A (1≦i≦n) and their reverse conversionfunctions G_(i): A×A→A, for a domain A. For arbitrary xεA, yεA, theconditions of:

F _(i)(G _(i)(x,y),y)=x;

G _(i)(F _(i)(x,y),y)=x

should be satisfied.

A single-term arithmetic operation ⋆: A^(n)→A^(n) and its reversesingle-term arithmetic operation ⊚: A^(n)→A^(n) are adopted below. Interms of these arithmetic operations, for arbitrary zεA^(n), thefollowing conditions of:

⋆(⊚z)=z;

⊚(⋆z)=z;

should be satisfied.

Particularly, in the following explanations, in the case where “A”represents a “t”-number bit space and “zεA^(n)” corresponds to a bitstring having “tn” bits in length, in the single-term arithmeticoperation ∘, bits in the bit string are cyclically shifted by apredetermined number of bits in a predetermined direction. After this,the resultant bit string is set to correspond to A^(n), therebyobtaining a result of the single-term arithmetic operation.

In the following description, “ceil (•)” represents that decimals shouldbe rounded off to the next whole number in “M” number system, and “floor(•)” represents that decimals should be cut off in “M” number system.

In the following embodiments, at least one of conversion functions F_(i)defined by positive integers M, s, and should satisfy the followingconditions of:

F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b);

F _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)),

for an arbitrary integer parameter b (1≦b≦M^(s)). This at least oneconversion function corresponds to the above-described Masuda-Aiharamapping with a parameter.

In the following embodiments, instead of the above-described conversionfunctions Ft, there can be employed a function (reverse mapping of theabove-described Masuda-Aihara mapping with a parameter) which is definedby positive integers M, s, and satisfies, for an arbitrary integerparameter b (1≦b≦M^(s)), the following conditions of:

F _(i)(y,b)=x ₁(q<x ₁);

F _(i)(y,b)=x ₂(x≦q),

where

x ₁=floor(M ^(−s) by);

x ₂=ceil((M ^(−s) b−1)y+M ^(s));

q=b(x ₂ −M ^(s))/(b−M ^(s)),

Fourth Embodiment

FIG. 7 is an exemplary diagram showing the schematic structure of aconverter according to the fourth embodiment of the present invention.

A converter 701 uses a predetermined parameter aεA. The converter 701includes a generating unit 702, a data accepting unit 703, a repetitioncontroller 704, and a converter 705.

The generating unit 702 accepts generated inputs, x₁, x₂, . . . ,x_(n)εA, whose length is “n” in total, and outputs generated outputs,y₁, y₂, . . . , y_(n)εA, whose length is “n” in total, using thefollowing recurrence formulas:

y ₁ =F ₁(x ₁ ,a);

y _(i+1) =F ^(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1);

The data accepting unit 703 accepts data inputs, k₁, k₂, . . . ,k_(n)εA, whose length is “n” in total, and gives the accepted datainputs to the generating unit 702.

The repetition controller 704 gives back the generating unit 702 thegenerated outputs sent from the generating unit 102 as generated inputs,repeatedly for an “m” (m≧0) number of times. In this case, the generatedoutput to be given at the end is a random number string, r₁, r₂, . . . ,r_(n)εA, whose length is “n” in total.

The converting unit 705 applies a single-term arithmetic operation ⋆ tothe random number string, r₁, r₂, . . . , r_(n)εA, to perform its dataconversion, that is,

(e ₁ ,e ₂ , . . . , e _(n))=(r ₁ ,r ₂ , . . . , r _(n)),

so as to output data outputs, e₁, e₂, . . . , e_(n), whose length is “n”in total.

The arithmetic operation can be accomplished at high speed by a parallelcomputer having a pipeline process function, and can be accomplishedalso by a general serial computer.

FIG. 8 is a flowchart for explaining a conversion process which iscarried out by a serial computer serving as the converter 701.

The converter 701 accepts data inputs, k₁, k₂, . . . , k_(n)εA, whoselength is “n” in total (Step S801).

The converter 701 substitutes the accepted data inputs respectively forx₁, x₂, . . . , x_(n)εA (Step S802).

After this, the converter 701 substitutes a value “m” for the countervariable “c” (Step S803).

Then, the converter 701 calculates the variables y₁, y₂, . . . , y_(n)εA(Step S804), using the recurrence formulas:

y ₁ =F ₁(x ₁ ,a),

y _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1).

The converter 701 checks whether the counter variable “c” is 0 (StepS805). In the case where it is determined that the counter variable “c”is not 0 (Step S805; No), the converter 701 substitutes the variables,y₁, y₂, . . . , y_(n), respectively for the variables x₁, x₂, . . . ,x_(n) (Step S806), and decrements the counter variable “c” by one (StepS807), and the flow returns to the procedure of the step S804.

In the case where the counter variable “c” is 0 (Step S805; Yes), theconverter 701 substitutes the variables y₁, y₂, . . . , y_(n)respectively for the variables r₁, r₂, . . . , r_(n)εA (Step S808).

The converter 701 uses a single-term arithmetic operation ⋆ for thevariables, r₁, r₂, . . . , r_(n)εA, to perform its data conversion, thatis,

(e ₁ ,e ₂ , . . . , e _(n))=⋆(r ₁ ,r ₂ , . . . , r _(n)).

Finally, the converter 701 outputs the variables, e₁, e₂, . . . , e_(n)(Step S810).

Fifth Embodiment

FIG. 9 is an exemplary diagram showing the schematic structure of aconverter which is in a pair relationship with the above-describedconverter 701.

A converter 901 according to the fifth embodiment of the presentinvention use the same arithmetic operations, functions, parameters,like “F_(i)”, “G_(i)”, ⊚, ⋆, “a”, “m”, as those used by the converter701.

The converter 901 uses a parameter “a”. The converter 901 includes agenerating unit 902, a data accepting unit 903, a converting unit 904,and a repetition controller 905.

The generating unit 902 accepts generated inputs, x₁, x₂, . . . ,x_(n)εA, whose length is “n” in total, and outputs generated outputs,y₁, y₂, . . . , Y_(n)εA whose length is “n” in total, using thefollowing recurrence formulas:

y ₁ =G ₁(x ₁ ,a);

y _(i+1) =G _(i+1)(x _(i+1) ,x _(i))(1≦i≦n−1).

The data accepting unit 903 accepts data inputs, h₁, h₂, . . . ,h_(n)εA.

The converting unit 905 uses a single-term arithmetic operation ⊚ forthe data inputs, h₁, h₂, . . . , h_(n)εA, to perform its dataconversion, that is,

(v ₁ ,v ₂ , . . . , v _(n))=⊚(h ₁ ,h ₂ , . . . , h _(n)), and

gives the generating unit 902 the results (v₁, v₂, . . . , v_(n)) of theconversion.

The repetition controller 905 gives back the generating unit 902 thegenerated outputs sent from the generating unit 902 as generated inputs,repeatedly for an “m” (m≧0) number of times. In this case, the generatedoutputs to be given at the end are data outputs, s₁, s₂, . . . ,s_(n)εA, whose length is “n” in total.

This calculation (data conversion) can be executed at high speed by aparallel computer having a pipeline process function. However, the abovecalculation may be executed by a generally-used serial computer.

FIG. 10 is a flowchart for explaining a conversion process which iscarried out by a serial computer serving as the converter 901.

The converter 901 accepts data inputs, h₁, h₂, . . . , h_(n)εA, whoselength is “n” in total (Step S1001).

The converter 901 uses a single-term arithmetic operation ⊚ for the datainputs, h₁, h₂, . . . , h_(n), so as to perform data conversion (StepS1002), that is,

(v ₁ ,v ₂ , . . . , v _(n))=⊚(h ₁ ,h ₂ , . . . , h _(n)).

The converter 901 substitutes variables, v₁, v₂, . . . , v_(n)respectively for x₁, x₂, . . . , x_(n)εA (Step S1003).

The converter 901 substitutes a value “m” for the counter variable “C”(Step S1004).

Further, the converter 901 calculates the variables, y₁, y₂, . . . ,y_(n)εA (Step S1005), using the recurrence formulas:

y ₁ =G ₁(x ₁ ,a);

y _(i+1) =G _(i+1)(x _(i+1) ,x _(i))(1≦i≦n−1).

The converter 901 checks whether the counter variable “C” is 0 (StepS1006). In the case where it is determined that the counter variable “C”is not 0 (Step S1006; No), the converter 901 substitutes the variables,y₁, y₂, . . . , y_(n) respectively for the variables, x₁, x₂, . . . ,x_(n) (Step S1007), and decrements the counter value “C” by one (StepS1008), and the flow returns to the procedure of the step S1005.

On the contrary, in the case where it is determined that the countervariable “C” is 0 (Step S1006; Yes), the converter 901 substitutes thevariables, y₁, y₂, . . . , y_(n) respectively for the variables, s₁, s₂,. . . , s_(n)εA (Step S109).

Finally, the converter 901 outputs the variables, s₁, s₂, . . . , s_(n)(Step S1010).

Sixth Embodiment

Explanations will now be made to an encryption/decryption systemincluding the above-described converters 701 and 901 which are in a pairrelationship with each other. Either the converter 701 or the converter901 is used as an encrypting unit, and the other one is used as adecrypting unit, so that there are two different types of systems inaccordance with the combination of the two.

FIG. 11 is an exemplary diagram showing the schematic structure of theencryption/decryption system including both the converter 701 as theencrypting unit and the converter 901 as the decrypting unit.

An encryption/decryption system 1101 according to the sixth embodimentof the present invention includes an encrypting unit 1102 and adecrypting unit 1103. The encrypting unit 1102 includes theabove-described converter 701, while the decrypting unit 1103 includesthe converter 901 which is in a pair relationship with the converter701.

The encrypting unit 1103 accepts original data, as data inputs, k₁, k₂,. . . , k_(n), whose length is “n” in total, and outputs data outputs,e₁, e₂, . . . , e_(n), whose length is “n” in total, as encrypted data.

The decrypting unit 1104 accepts the encrypted data whose length is “in”in total, as data inputs, h₁, h₂, . . . , h_(n), and outputs dataoutputs, s₁, s₂, . . . , s_(n), whose length is “n” in total, asdecrypted data.

According to this structure, the vector-stream private key encryptingsystem can be realized.

FIG. 12 is an exemplary diagram showing the schematic structure of anencrypting/decryption system 1201, including the converter 901 servingas an encrypting unit and the converter 701 serving as a decryptingunit.

The encrypting/decryption system 1201 includes an encrypting unit 1202and a decrypting unit 1203. The encrypting unit 1202 includes theabove-described converter 901, while the decrypting unit 1203 includesthe converter 701 which is in a pair relationship with the converter901.

The encrypting unit 1202 accepts original data, as data inputs, h₁, h₂,. . . , h_(n), whose length is “n” in total, and outputs data outputss₁, s₂, . . . , s_(n) whose length is “n” in total, as encrypted data.

The decrypting unit 1203 accepts the encrypted data whose length is “n”as data inputs, k₁, k₂, . . . , k_(n), and outputs data outputs, e₁, e₂,. . . , e_(n), whose length is “n” in total as decrypted data.

Likewise the above, according to this embodiment as well, avector-stream private key encrypting system can be realized.

The single-term arithmetic operations ⊚ and ⋆ adopted in the fourth tosixth embodiments of the present invention will now exemplarilydescribed. In the case where “A” represents one bit space and “zεA^(n)”corresponds to a bit string having “n” bits in length, in thesingle-term arithmetic operation ⊚, the following specific calculationcan be employed

⊚(z ₁ ,z ₂ , . . . , z _(a−1) ,z _(a) , . . . , z _(n))=(z _(a) , . . ., z _(n) ,z ₁ ,z ₂ , . . . , z _(n−1)).

This is an “a−1” bit(s) circulation (cyclical shift) arithmeticoperation (can also be called “n−a+1” bit(s) circulation arithmeticoperation). In terms of the arithmetic operation ⋆, there can beemployed the opposite bit circulation arithmetic operation for shiftingbits in the bit string in the opposite direction to that in the case ofthe arithmetic operation ⊚. An example of this is

⋆(z _(a) , . . . , z _(n) , . . . , z ₁ ,z ₂ , . . . , z _(a−1))=(z ₁ ,z₂ , . . . , z _(a−1) ,z _(a) , . . . , z _(n))

Even in the case where A is t (t>1), such a bit circulation arithmeticoperation can spontaneously be expanded, and can be adopted for thepresent invention.

In the following explanations, a multi-stage converter, including theconverter 701 and the converter 901 in multi-stages which are in a pairrelation with each other, will be described in each of the seventh andeighth embodiments. Further, an encryption/decryption system using theabove multi-stage converter will be described in each of the ninth andtenth embodiments.

Seventh Embodiment

FIG. 13 is an exemplary diagram showing the schematic structure of amulti-stage converter 1301 according to the seventh embodiment of thepresent invention.

The multi-stage converter 1301 includes a “u” number of converters 701(the “J”-th converter is called M_(j) (1≦j≦u)) and a multi-stagekey-input accepting unit 1302.

The multi-stage key-input accepting unit 1302 accepts parameter inputs,a₁, a₂, . . . , a_(u)εA, whose length is “u” in total. The multi-stagekey-input accepting unit 1302 sets the “j”-th parameter input a_(j) as apredetermined parameter of the corresponding converter 701 M_(j).

The converter 701 M_(i) accepts multi-stage conversion inputs, k₁, k₂, .. . , k_(n), as data inputs.

Those data outputs which are output by the converter 701 M_(i) (1≦i≦u−1)are given to the converter 701 M_(i+1), as data inputs.

The converter 701 M_(u) outputs data outputs, e₁, e₂, . . . , e_(n),whose length is “n” in total, as multi-stage conversion outputs.

Eighth Embodiment

FIG. 14 is an exemplary diagram showing the schematic structure of amulti-stage converter 1401 which is in a pair relationship with theabove-described multi-stage converter 1301.

The multi-stage converter 1401 includes a “u” number of converters 901(the “j”-th converter is called N_(j) (1≦j≦u)), and a multi-stagekey-input accepting unit 1402.

The multi-stage key-input accepting unit 1402 accepts parameter inputsa₁, a₂, . . . , a_(n)εA whose length is “n” in total. The multi-stagekey-input accepting unit 1402 sets the “J”-th parameter input a_(j) as apredetermined parameter of the corresponding converter 901 N_(j).

The converter 901 N_(u) accepts multi-stage conversion inputs, h₁, h₂, .. . , h_(n), whose length is “n” in total, as data inputs.

Those data outputs which are output by the converter 901 N_(i+1)(1≦i≦u−1) are given to the converter 901 N₁, as data inputs.

The converter 901 N₁ outputs data outputs, s₁, s₂, . . . , s_(n), whoselength is “n” in total, as multi-stage conversion outputs.

Ninth Embodiment

FIG. 15 is an exemplary diagram showing the schematic structure of anencryption/decryption system 1501 including the above-describedmulti-stage converter 1301 and the multi-stage converter 1401 which arein a pair relationship with each other.

The encryption/decryption system 1501 includes the above-describedmulti-stage converter 1301, serving as an encrypting unit 1502, and theabove-described multi-stage converter 1401, serving as a decrypting unit1503.

F_(i), G_(i), ⋆ and ⊚ are commonly used by the encrypting unit 1502 andthe decrypting unit 1503.

Those parameter inputs, a₁, a₂, . . . , a_(u), are commonly accepted bythe encrypting unit 1502 and the decrypting unit 1503.

The encrypting unit 1502 accepts original data as multi-stage conversioninputs, k₁, k₂, . . . , k_(n), whose length is “n” in total, and outputsmulti-stage conversion outputs, e₁, e₂, . . . , e_(n), whose length is“n” in total as encrypted data.

The decrypting unit 1503 accepts the encrypted data as multi-stageconversion inputs, h₁, h₂, . . . , h_(n), whose length is “n” in total,and outputs multi-stage conversion outputs, s₁, s₂, . . . , s_(n), whoselength is “n” in total as decrypted data.

According to this embodiment, a vector-stream private key encryptionsystem can be realized.

Tenth Embodiment

FIG. 16 is an exemplary diagram showing the schematic structure of anencryption/decryption system 1601, including the above-describedmulti-stage converter 1301 and the multi-stage converter 1401 which arein a pair relationship with each other.

The encryption/decryption system 1601 includes the above-describedmulti-stage converter 1401 as an encrypting unit 1602 and theabove-described multi-stage converter 1301 as a decrypting unit 1603.

F_(i), G_(i), ⋆ and ⊚ are commonly used by the encrypting unit 1602 andthe decrypting unit 1603.

Those parameter inputs, a₁, a₂, . . . , a_(u), are commonly accepted bythe encrypting unit 1602 and the decrypting unit 1603.

The encrypting unit 1602 accepts original data as multi-stage conversioninputs, h₁, h₂, . . . , h_(n), whose length is “n” in total, and outputsmulti-stage conversion outputs, s₁, s₂, . . . , s_(n), whose length is“n” in total as encrypted data.

Further, the decrypting unit 1603 accepts the encrypted data asmulti-stage conversion inputs, k₁, k₂, . . . , k_(n), whose length is“n” in total, and outputs multi-stage conversion outputs, e₁, e₂, . . ., e_(n), whose length is “n” in total as the decrypted data.

According to this embodiment also, a vector-stream private keyencryption system can be realized.

In the vector-stream private key encryption system, the computationparallelism thereof is enhanced, if the dimension number “n” is setlarge. Hence, with the utilization of an FPGA (Field Programmable GateArray), etc. or with the structure suitable for parallel processingusing a dedicated chip, etc., high-speed processing may be furtherexpected.

Eleventh Embodiment

Likewise the disclosure of Patent No. 3030341 and Unexamined JapanesePatent Application KOKAI Publication No. 2001-175168, when the basicconversion of the present invention has an equal distribution, it alsoresults in an equal distribution of the multi-dimensional vector(s) inthe synthetic conversion of the present conversion.

FIG. 17 is shows a data distribution of data generated by athree-dimensional vector-stream private key encryption system, in a cube[0, 1]³.

As seen from FIG. 17, it is obvious that data is equally distributed inthe cube.

In the encryption process, the statistical stability, like an equalfrequency characteristic, is required. As obvious from FIG. 17,according to the technique of the present invention, the datadistribution shows the equal frequency characteristic.

The system of the present invention can be realized by a generalcomputer, without the need for a dedicated system. A program and datafor controlling a computer to execute the above-described processes maybe recorded on a medium (a floppy disk, CD-ROM, DVD or the like) anddistributed, and the program may be installed into the computer and runon an OS (Operating System) to execute the above-described processes,thereby achieving the system of the present invention. The above programand data may be stored in a disk device or the like in the server deviceon the Internet, and embedded in a carrier wave. The program and dataembedded in the carrier wave may be downloaded into the computer so asto realize the system of the present invention.

Various embodiments and changes may be made thereonto without departingfrom the broad spirit and scope of the invention. The above-describedembodiments are intended to illustrate the present invention, not tolimit the scope of the present invention. The scope of the presentinvention is shown by the attached claims rather than the embodiments.Various modifications made within the meaning of an equivalent of theclaims of the invention and within the claims are to be regarded to bein the scope of the present invention.

This application is based on Japanese Patent Application No. 2001-261698filed on Aug. 30, 2001, and including specification, claims, drawingsand summary. The disclosure of the above Japanese Patent Application isincorporated herein by reference in its entirety.

1. A converter using: an “n” (n≧1) number of conversion functions,F_(i): A×A→A (1≦i≦n), with respect to a domain A; a binary arithmeticoperation, ⋆: A×A→A, and its reverse binary arithmetic operation, ⊚:A×A→A, wherein, for arbitrary xεA, yεA, conditions of(x⋆y)⊚y=x, and(x⊚y)⋆y=x are satisfied; and a predetermined parameter, aεA, and saidconverter comprising a generating unit, a key accepting unit, arepetition controller, a data accepting unit, and a converting unit, andwherein: said generating unit accepts generated inputs, x₁, x₂, . . . ,x_(n)εA, whose length is “n” in total, and generates generated outputs,y₁, y₂, . . . , y_(n)εA, whose length is “n” in total using recurrenceformulasy ₁ =F ₁(x ₁ ,a), andy _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1); said key accepting unitaccepts key inputs, k₁, k₂, . . . , k_(n)εA, whose length is “n” intotal, and gives the accepted key inputs as generated inputs to saidgenerating unit; said repetition controller gives the generated outputsfrom said generating unit as generated inputs to said generating unit,for an “m” (m≧0) number of times, and sets one of the generated outputsto be given at end as a random number string, r₁, r₂, . . . , r_(n)εA,whose length is “n” in total; said data accepting unit accepts datainputs, d₁, d₂, . . . , d_(n)εA, whose length is “n” in total; and saidconverting unit converts data for any integers “i” in a range between 1and “n” using a formulae_(i)=d_(i)⋆r_(i), and outputs data outputs, e₁, e₂, . . . , e_(n)εA,whose length is “n” in total.
 2. The converter according to claim 1,wherein each of the binary arithmetic operations ⊚ and ⋆ is exclusiveOR.
 3. The converter according to claim 1, wherein at least one of theconversion functions F_(i) defined by positive integers M, s, andsatisfies following conditions, for an arbitrary integer parameter b(1≦b≦M^(s)),F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b),andF _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)), in caseswhere: “ceil (•)” represents that decimals should be rounded off to anext whole number in “M” number system; and “floor (•)” represents thatdecimals should be cut off in “M” number system.
 4. The converteraccording to claim 1, wherein at least one of the conversion functionsF_(i) defined by positive integers M, s, and satisfies followingconditions, for an arbitrary integer parameter, b (1≦b≦M^(s)),F _(i)(y,b)=x ₁(q<x ₁);F _(i)(y,b)=x ₂(x ₁ ≦q), wherex ₁=floor(M ^(−s) by);x ₂=ceil((M ^(−s) b−1)y+M ^(s));q=b(x ₂ −M ^(s))/(b−M ^(s)), in cases where: “ceil (•)” represents thatdecimals should be rounded off to a next whole number in “M” numbersystem; and “floor (•)” represents that decimals should be cut off in“M” number system.
 5. A converting method using: an “n” (n≧1) number ofconversion functions, F_(i): A×A→A (1≦i≦n), with respect to a domain A;a binary arithmetic operation, ⋆: A×A→A, and its reverse binaryarithmetic operation, ⊚: A×A→A, wherein, for arbitrary xεA, yεA,conditions of(x⋆y)⊚y=x, and(x⊚y)⋆y=x are satisfied; and a predetermined parameter, aεA, and saidconverting method comprising a generating step, a key accepting step, arepetition controlling step, a data accepting step, and a convertingstep, and wherein: said generating step includes accepting generatedinputs, x₁, x₂, . . . , x_(n)εA, whose length is “n” in total, andgenerating generated outputs, y₁, y₂, . . . , y_(n)εA, whose length is“n” in total using recurrence formulas,y ₁ =F ₁(x ₁ ,a), andy _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1); said key accepting stepincludes accepting key inputs, k₁, k₂, . . . , k_(n)εA, whose length is“n” in total, and giving the accepted key inputs as generated inputs tosaid generating step; said repetition controlling step includes givingthe generated outputs from said generating step as generated inputs tosaid generating step, for an “m” (m≧0) number of times, and setting oneof the generated outputs to be given at end as a random number string,r₁, r₂, . . . , r_(n)εA, whose length is “n” in total; said dataaccepting step includes accepting data inputs, d₁, d₂, . . . , d_(n)εA,whose length is “n” in total; and said converting step includesconverting data for any integers “i” in a range between 1 and “n” usinga formula,e_(i)=d_(i)⋆r_(i), and outputting data outputs, e₁, e₂, . . . , e_(n)εA,whose length is “n” in total.
 6. A converting method using: an “n” (n≧1)number of conversion functions, F_(i): A×A→A (1≦i≦n), with respect to adomain A; a binary arithmetic operation, ⋆: A×A→A, and its reversebinary arithmetic operation, ⊚: A×A→A, wherein, for arbitrary xεA, yεA,conditions of(x⋆y)⊚y=x, and(x⊚y)⋆y=x are satisfied; and a predetermined parameter, aεA, and saidconverting method comprising a generating step, a key accepting step, arepetition controlling step, a data accepting step, and a convertingstep, and wherein: said generating step includes accepting generatedinputs, x₁, x₂, . . . , x_(n)εA, whose length is “n” in total, andgenerating generated outputs, y₁, y₂, . . . , y_(n)εA, whose length is“n” in total using recurrence formulas,y ₁ =F ₁(x ₁ ,a), andy _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1); said key accepting stepincludes accepting key inputs, k₁, k₂, . . . , k_(n)εA whose length is“n” in total, and giving the accepted key inputs as generated inputs tosaid generating step; said repetition controlling step includes givingthe generated outputs from said generating step as generated inputs tosaid generating step, for an “m” (m≧0) number of times, and setting oneof the generated outputs to be given at end as a random number string,r₁, r₂, . . . , r_(n)εA, whose length is “n” in total; said dataaccepting step includes accepting data inputs, d₁, d₂, . . . , d_(n)εA,whose length is “n” in total; and said converting step includesconverting data for any integers “i” in a range between 1 and “n” usinga formulae_(i)=d_(i)⋆r_(i), and outputting data outputs, e₁, e₂, . . . , e_(n)εA,whose length is “n” in total.
 7. The converting method according toclaim 5, wherein each of the binary arithmetic operations ⊚ and ⋆ isexclusive OR.
 8. The converting method according to claim 5, wherein atleast one of the conversion functions F_(i) defined by positive integersM, s, and satisfies following conditions, for an arbitrary integerparameter b (1≦b≦M^(s)),F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b), andF _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)), in caseswhere: “ceil (•)” represents that decimals should be rounded off to anext whole number in “M” number system; and “floor (•)” represents thatdecimals should be cut off in “M” number system.
 9. The convertingmethod according to claim 5, wherein at least one of the conversionfunctions F_(i) defined by positive integers M, s, and satisfiesfollowing conditions, for an arbitrary integer parameter, b (1≦b≦M^(s)),F _(i)(y,b)=x ₁(q<x ₁);F _(i)(y,b)=x ₂(x ₁ ≦q), wherex ₁=floor(M ^(−s) by);x ₂=ceil((M ^(−s) b−1)y+M ^(s));q=b(x ₂ −M ^(s))/(b−M ^(s)), in cases where: “ceil (•)” represents thatdecimals should be rounded off to a next whole number in “M” numbersystem; and “floor (•)” represents that decimals should be cut off in“M” number system.
 10. A converting method using: an “n” (n≧1) number ofconversion functions, F_(i): A×A→A, (1≦i≦n) and their reverse conversionfunctions, G_(i): A×A→A, with respect to a domain A, wherein, forarbitrary xεA and yεA, conditions ofF _(i)(G _(i)(x,y),y)=x, andG _(i)(F _(i)(x,y),y)=x, are satisfied; a binary arithmetic operation,⋆: A^(n)→A^(n), and its reverse binary arithmetic operation, ⊚:A^(n)→A^(n), wherein, for arbitrary zεA^(n), conditions of⋆(⊚z)=z,and⊚(⋆z)=z are satisfied; and a predetermined parameter, aεA, and saidconverting method comprising a generating step, a data accepting step, arepetition controlling step, and a converting step, and wherein: saidgenerating step includes accepting generated inputs, x₁, x₂, . . . ,x_(n)εA, whose length is “n” in total, and generating generated outputs,y₁, y₂, . . . , y_(n)εA, whose length is “n” in total using recurrenceformulas,y ₁ =F ₁(x ₁ ,a), andy _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1); said data accepting stepincludes accepting data inputs, k₁, k₂, . . . , k_(n)εA, whose length is“n” in total, and giving the accepted data inputs as generated inputs tosaid generating step; said repetition controlling step includes givingthe generated outputs from said generating step as generated inputs tosaid generating step, for an “m” (m≧0) number of times, and setting oneof the generated outputs to be given at end as a random number string,r₁, r₂, . . . , r_(n)εA, whose length is “n” in total; and saidconverting step includes applying a single-term arithmetic operation, A,to the random number string, r₁, r₂, . . . , r_(n)εA, to perform itsdata conversion, that is,(e ₁ ,e ₂ , . . . , e _(n))=⋆(r ₁ ,r ₂ , . . . , r _(n)) and outputtingdata outputs, e₁, e₂, . . . , e_(n), whose length is “n” in total.
 11. Aconverting method using: an “n” (n≧1) number of conversion functions,F_(i): A×A→A (1≦i≦n), and their reverse conversion functions, G_(i):A×A→A, with respect to a domain A, wherein, for arbitrary xεA and yεA,conditions ofF _(i)(G _(i)(x,y),y)=x, andG _(i)(F _(i)(x,y),y)=x, are satisfied; a binary arithmetic operation,⋆: A^(n)→A^(n), and its reverse binary arithmetic operation, ⊚:A^(n)→A^(n), wherein, for arbitrary zεA^(n), conditions of⋆(⊚z)=z, and⊚(⋆z)=z are satisfied; and a predetermined parameter, aεA, and saidconverting method comprising a generating step, a data accepting step, aconverting step, and a repetition controlling step, and wherein: saidgenerating step includes accepting generated inputs, x₁, x₂, . . . ,x_(n)εA, whose length is “n” in total, and generating generated outputs,y₁, y₂, . . . , y_(n)εA, whose length is “n” in total using recurrenceformulas,y ₁ =G ₁(x ₁ ,a), andy _(i+1) =G _(i+1)(x _(i+1) x _(i))(1≦i≦n−1); said data accepting stepincludes accepting data inputs, h₁, h₂, . . . , h_(n)εA, whose length is“n” in total; said converting step includes applying a single-termarithmetic operation, ⋆, to the data inputs, h₁, h₂, . . . , h_(n), toperform its data conversion, that is,(v ₁ ,v ₂ , . . . , v _(n))=⋆(h ₁ ,h ₂ , . . . , h _(n)), and givingresults of the data conversion, v₁, v₂, . . . , v_(n), to saidgenerating step; and said repetition controlling step includes givingthe generated outputs from said generating step as generated inputs tosaid generating step, for an “m” (m≧0) number of times, and setting oneof the generated outputs to be given at end as data outputs, s₁, s₂, . .. , s_(n)εA, whose length is “n” in total.
 12. The converting methodaccording to claim 10, wherein in cases where “A” represents a“t”-number bit space, and “zεA^(n)” corresponds to a bit string having“tn” bits in length, in the single-term arithmetic operation ⊚, bits inthe bit string are shifted by a predetermined number of bits in apredetermined direction, and its resultant bit string is set tocorrespond to A^(n), thereby obtaining a result of the single-termarithmetic operation ⊚.
 13. The converting method according to claim 10,wherein at least one of the conversion functions F_(i) defined bypositive integers M, s, and satisfies following conditions, for anarbitrary integer parameter b (1≦b≦M^(s)),F _(i)(x,b)=ceil(xM ^(s) /b)(1≦x≦b), andF _(i)(x,b)=floor(M ^(s)(x−b)/(M ^(s) −b))+1(b≦x≦M ^(s)), in caseswhere: “ceil (•)” represents that decimals should be rounded off to anext whole number in “M” number system; and “floor (•)” represents thatdecimals should be cut off in “M” number system.
 14. The convertingmethod according to claim 10, wherein at least one of the conversionfunctions F_(i) defined by positive integers M, s, and satisfiesfollowing conditions, for an arbitrary integer parameter, b (≦b≦M^(s)),F _(i)(y,b)=x ₁(q<x ₁);F _(i)(y,b)=x ₂(x ₁ ≦q), wherex ₁=floor(M ^(−s) by);x ₂=ceil((M ^(−s) b−1)y+M ^(s));q=b(x ₂ −M ^(s))/(b−M ^(s)), in cases where: “ceil (•)” represents thatdecimals should be rounded off to a next whole number in “M” numbersystem; and “floor (•)” represents that decimals should be cut off in“M” number system.
 15. A multi-stage converting method comprising: a “u”number of converting steps (a “j”-th converting step is called aconverting step M_(j) (1≦j≦u)) of using the converting method accordingto claim 10; and a multi-stage key-input accepting step of acceptingparameter inputs a₁, a₂, . . . , a_(u)εA whose length is “n” in total,and setting a “j”-th parameter input, a_(j), included in the acceptedparameter inputs, as a predetermined parameter “a” of the convertingstep M_(j), and wherein a converting step M₁ included in said “u” numberof converting steps includes accepting multi-stage conversion inputs,k₁, k₂, . . . , k_(n), whose length is “n” in total, as data inputs,data outputs, which are output at a converting step M_(i) (1≦i≦u−1)included in said “u” number of converting steps, are given to aconverting step M_(i+1) included in said “u” number of converting steps,as data inputs, and a converting step M_(u) included in said “u” numberof converting steps includes outputting data outputs, e₁, e₂, . . . ,e_(n), whose length is “n” in total, as multi-stage conversion outputs.16. A multi-stage converting method comprising: a “u” number ofconverting steps (a “j”-th converting step is called a converting stepM_(j) (1≦j≦u)) of using the converting method according to claim 11; anda multi-stage key-input accepting step of accepting parameter inputs a₁,a₂, . . . , a_(u)εA whose length is “n” in total, and setting a “j”-thparameter input, a_(j), included in the accepted parameter inputs, as apredetermined parameter “a” of the converting step M_(j), and wherein aconverting step M_(u) included in said “u” number of converting stepsincludes accepting multi-stage conversion inputs, h₁, h₂, . . . , h_(n),whose length is “n” in total, as data inputs, data outputs, which areoutput at a converting step M_(i+1) (1≦i≦u−1) included in said “u”number of converting steps, are given to a converting step M_(i)included in said “u” number of converting steps, as data inputs, and aconverting steps M₁ included in said “u” number of converting stepsincludes outputting data outputs, s₁, s₂, . . . , s_(n), whose length is“n” in total, as multi-stage conversion outputs.
 17. The converteraccording to claim 1, wherein at least one of the conversion functionsF_(i) defined by positive integers M, s, is a second degree polynomialin x over module M^(s), and satisfies, for an arbitrary integerparameter b (1≦b≦M^(s)) and a predefined function g of b, the followingconditions of:F _(i)(x,b)=2x(x+g(b))mod M ^(s).
 18. A converter using: an “n” (n≧1)number of conversion functions F_(i): A×A→A (1≦i≦n) and their reverseconversion functions G_(i): A×A→A, with respect to a domain A, wherein,for arbitrary xεA and yεA, conditions ofF _(i)(G _(i)(x,y),y)=x, andG _(i)(F _(i)(x,y),y)=x, are satisfied; a binary arithmetic operation,⋆: A^(n)→A^(n), and its reverse binary arithmetic operation, ⊚:A^(n)→A^(n), wherein, for arbitrary zεA^(n), conditions of⋆(⊚z)=z, and⊚(⋆z)=z are satisfied; and a predetermined parameter, aεA, and saidconverter comprising a generating unit, a data accepting unit, arepetition controller, and a converting unit, and wherein: saidgenerating unit accepts generated inputs, x₁, x₂, . . . , x, εA, whoselength is “n” in total, and generates generated outputs, y₁, y₂, . . . ,x_(n)εA, whose length is “n” in total using recurrence formulasy ₁ =F ₁(x ₁ ,a), andy _(i+1) =F _(i+1)(x _(i+1) ,y _(i))(1≦i≦n−1); said data accepting unitaccepts data inputs, k₁, k₂, . . . , k_(n)εA, whose length is “n” intotal, and gives the accepted data inputs as generated inputs to saidgenerating unit; said repetition controller gives the generated outputsfrom said generating unit as generated inputs to said generating unit,for an “m” (m≧0) number of times, and sets one of the generated outputsto be given at end as a random number string, r₁, r₂, . . . , r_(n)εA,whose length is “n” in total; said converting unit applies a single-termarithmetic operation, ⋆, to the random number string, r₁, r₂, . . . ,r_(n)εA, to perform its data conversion, that is,(e ₁ ,e ₂ , . . . , e _(n))=⋆(r ₁ ,r ₂ , . . . , r _(n)), and outputsdata outputs, e₁, e₂, . . . , e_(n), whose length is “n” in total; andwherein at least one of the conversion functions F_(i) defined bypositive integers M, s, is a second degree polynomial in x over moduleM^(s), and satisfies, for an arbitrary integer parameter b (1≦b≦M^(s))and a predefined function g of b, the following conditions of:F _(i)(x,b)=2x(x+g(b))mod M ^(s).
 19. The converting method according toclaim 5, wherein at least one of the conversion functions F_(i) definedby positive integers M, s, is a second degree polynomial in x overmodule M^(s), and satisfies, for an arbitrary integer parameter b(1≦b≦M^(s)) and a predefined function g of b, the following conditionsof:F _(i)(x,b)=2x(x+g(b))mod M ^(s).
 20. The converting method according toclaim 10, wherein at least one of the conversion functions F_(i) definedby positive integers M, s, is a second degree polynomial in x overmodule M^(s), and satisfies, for an arbitrary integer parameter b(1≦b≦M^(s)) and a predefined function g of b, the following conditionsof:F _(i)(x,b)=2x(x+g(b))mod M ^(s).
 21. An information recording mediumstoring a program for controlling a computer to serve as the converteras recited in claim 1.